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Symmetry Restoration and the Gluon Mass in the Landau Gauge Abstract SciPost Phys. 10, 035 (2021) Symmetry restoration and the gluon mass in the Landau gauge Urko Reinosa1, Julien Serreau2, Rodrigo Carmo Terin3,4 and Matthieu Tissier4 1 Centre de Physique Théorique (CPHT), CNRS, École Polytechnique, Institut Polytechnique de Paris, Route de Saclay, F-91128 Palaiseau, France. 2 Université de Paris, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France. 3 Universidade do Estado do Rio de Janeiro (UERJ), Instituto de Física, Departamento de Física Teórica, Rua São Francisco Xavier 524, Maracanã, Rio de Janeiro, Brasil, CEP 20550-013 4 Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée, LPTMC, F-75005 Paris, France Abstract We investigate the generation of a gluon screening mass in Yang-Mills theory in the Landau gauge. We propose a gauge-fixing procedure where the Gribov ambiguity is overcome by summing over all Gribov copies with some weight function. This can be formulated in terms of a local field theory involving constrained, nonlinear sigma model fields. We show that a phenomenon of radiative symmetry restoration occurs in this the- ory, similar to what happens in the standard nonlinear sigma model in two dimensions. This results in a nonzero gluon screening mass, as seen in lattice simulations. Copyright U. Reinosa et al. Received 28-04-2020 This work is licensed under the Creative Commons Accepted 20-01-2021 Check for Attribution 4.0 International License. Published 16-02-2021 updates Published by the SciPost Foundation. doi:10.21468/SciPostPhys.10.2.035 Contents 1 Introduction1 2 Gauge fixing2 3 Symmetry restoration5 4 Mass generation8 5 Conclusions 10 A Approximation scheme 11 References 12 1 SciPost Phys. 10, 035 (2021) 1 Introduction The similarities between the nonlinear sigma (NLσ) model in d = 2 spacetime dimensions and Quantum Chromodynamics (QCD) in d = 4 have been reported long ago. Both theories exhibit asymptotic freedom [1–4]. Moreover, their low-energy excitations are gapped, while their microscopic descriptions involve massless fields. In the case of the NLσ model, this ap- parent change of spectrum is a consequence of the radiative restoration of a symmetry as we now recall in the simplest O(N)/O(N 1) case. The NLσ model can be interpreted as de- scribing the ordered phase of a N component− vectorial model, where the radial fluctuations are frozen. The theory therefore involves N 1 massless Goldstone modes. However, the Coleman-Mermin-Wagner theorem [6,7 ] states− that no such ordered phase exists in d = 2. The true spectrum of the theory involves, instead, N degenerate massive modes, with an ex- ponentially small mass m µ exp( const.=g), with µ an ultraviolet (UV) scale and g the (running) coupling constant∼[5]. This− symmetry restoration phenomenon is best understood in the Wilson functional renormalization-group (RG) approach, where one follows the evolu- tion of the effective potential as modes above a RG scale k are progressively integrated out [8]. For k of the order of the microscopic scale of the theory, one starts with an O(N)-symmetric potential, strongly peaked around a nonzero value, which ensures that the radial modes are frozen and that only the transverse, pseudo-Goldstone degrees of freedom (d.o.f.) contribute. These massless modes lead to large infrared fluctuations which, in d = 2, are strong enough to result in an effective symmetry restoration: the minimum of the effective potential decreases with decreasing k and eventually vanishes below some scale kr . One can picture that regime as an incoherent collection of regions of broken symmetry of size 1=kr . On infrared scales, the symmetry in internal space is effectively restored and all modes are massive and degenerate. A somewhat similar situation occurs in Yang-Mills theories, where, despite the fact that the microscopic d.o.f. of the theory (the gluons) appear massless on UV scales, the long distance, physical excitations are massive glueballs. Not only that: it is, by now, well-established [9–12] that, in the Landau gauge, the gluon propagator actually reaches a finite nonzero value at vanishing momentum, corresponding to a nonzero screening mass. It is not clear at the mo- ment whether this results purely from the nontrivial infrared dynamics between the massless degrees of freedom of the Faddeev-Popov (FP) action, or if it originates from a deformation of the latter due to the gauge-fixing procedure on the lattice, in particular, the way the Gribov ambiguity is dealt with [13–15]. In Ref. [16], a gauge-fixing procedure was proposed in the Landau gauge, where Gribov copies are averaged over with a nontrivial weight, which provided an intriguing novel con- nection with the physics of the NLσ model described above. First, the continuum formulation of this procedure involves a set of auxiliary NLσ fields and, second, the existence of super- symmetries in the NLσ sector effectively reduces the number of spacetime dimensions by 2, a phenomenon similar to what happens (for sufficiently large number of dimensions) in disor- dered systems in statistical physics, known as the Parisi-Sourlas dimensional reduction [17]. Just like the Higgs’ phenomenon [18], the coupling of these constrained NLσ fields to the gluon gives a mass to the latter. The main point of Ref. [16] was to actually explore the pos- sibility to explain the observed gluon mass on the lattice in this way. However, the proposed scenario relied on a questionable inversion of limits and was, thus, not completely satisfactory. An attempt to circumvent that problem was proposed in Ref. [19] in an extension of the procedure of Ref. [16] to the class of nonlinear Curci-Ferrari-Delbourgo-Jarvis gauges. It was shown there that a gluon mass is indeed dynamically generated due to collective effects at large distances. Unfortunately, the latter tends to zero in the Landau gauge limit. In the present paper, we come back to a simpler setup by considering a slight deformation of the original proposal of Ref. [16], directly in the Landau gauge. We show, first, that there exists some 2 SciPost Phys. 10, 035 (2021) values of the gauge-fixing parameters for which the symmetry of the NLσ sector is radiatively restored and, second, that, whenever this happens, this results in a nonzero gluon mass at tree level. 2 Gauge fixing In this Section, we describe our gauge-fixing procedure and its formulation in terms of a local field theory with auxiliary fields. We concentrate on the Landau gauge, for which lots of information were obtained by lattice simulations. In practice, when a gauge configuration a a a Aµ = Aµ t (with t the generators of the group in the fundamental representation) has been selected, one has to find a gauge transformation U such that U @µAµ = 0 , (1) U † i † where the gauge transformation reads Aµ = UAµU + g U@µU . The gauge can be equivalently fixed by imposing that U extremizes Z 2 d U f [A, U] d x tr Aµ , (2) ≡ at fixed Aµ. As first stressed by Gribov [13], there exists in general many solutions Ui, called Gribov copies, to this problem. To characterize unambiguously the gauge-fixing procedure, it is necessary to supplement the condition given in Eq. (1) with a rule describing how to deal with these Gribov copies. The general strategy put forward in Ref. [16] is to sum over the different copies, with a nonuniform weight function P[A, U]. The gauge-fixing procedure applied to some operator O is then: P Ui i O[A ]P[A, Ui] O[A] P , (3) hh ii ≡ i P[A, Ui] where the sums run over all Gribov copies. As it should be for a bona fide gauge fixing, a gauge-invariant operator is not modified by this procedure, Oinv = Oinv, thanks to the denominator appearing in Eq. (3). This implies in particularhh thatii the average of a gauge- invariant observable does not depend on the particular choice of weight function P. In this article, we use Det F A, U ζ ( [ ] + 1) β f [A,U] P[A, U] e− , (4) ≡ Det(F[A, U]) j j where F[A, U] is the Hessian of the functional (2) on the group manifold at fixed A, U that is, the FP operator in the Landau gauge F[A, U]=Fˆ[A ], with ˆab ab acb c (d) F [A; x, y] @µ[@µδ + g f Aµ(x)]δ (x y). Here, β and ζ are two gauge-fixing parameters of≡ mass − dimension 2. For ζ = 0, the ratio− of functional determinants on the right- hand side reduces to the sign of the determinant of the FP operator and the weight (4) identifies with the one proposed in Ref. [16]. In this case, the resulting local field theory (see below) is of topological nature with the important consequence that the corresponding auxiliary fields give no loop contributions. A nonzero ζ relaxes those topological constraints and allows for a richer structure, in particular, the possibility of radiative symmetry restoration. It is also interesting to note that the gauge-fixing proposed here, although very differ- ent in spirit, shares some qualitative similarities with the renown (refined) Gribov-Zwanziger (GZ) approach [13, 14, 20, 21]. In the latter, the integral over gauge field configurations is 3 SciPost Phys. 10, 035 (2021) restricted to the first Gribov region—where the FP operator is positive definite—which even- tually strongly favors configurations near the first Gribov horizon—where the FP operator has its first zero eigenvalue. The present approach involves copies over all Gribov regions but the weight in (3) can be engineered so as to favor similar configurations as in the GZ approach. 1 With the choice (4), for ζ = 0, the numerator favors the copies near the Gribov horizons whereas the exponential factor6 suppresses the contribution from higher-than-the-first Gribov regions.
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